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Abstract

In many scientific and medical applications, such as laser systems and microscopes, wavefront-sensor-less (WFSless) adaptive optics (AO) systems are used to improve the laser beam quality or the image resolution by correcting the wavefront aberration in the optical path. The lack of direct wavefront measurement in WFSless AO systems imposes a challenge to achieve efficient aberration correction. This paper presents an aberration correction approach for WFSlss AO systems based on the model of the WFSless AO system and a small number of intensity measurements, where the model is identified from the input-output data of the WFSless AO system by black-box identification. This approach is validated in an experimental setup with 20 static aberrations having Kolmogorov spatial distributions. By correcting N = 9 Zernike modes (N is the number of aberration modes), an intensity improvement from 49% of the maximum value to 89% has been achieved in average based on N + 5 = 14 intensity measurements. With the worst initial intensity, an improvement from 17% of the maximum value to 86% has been achieved based on N + 4 = 13 intensity measurements.

Figures (7)

Schematic of a common closed-loop WFSless AO system. The incident light beam is disturbed in front of the entrance pupil. The control system adapts the control signal u(k) to maximize the intensity measurement y(k).

Cost function J(x̂) depends on the number of data points for solving the NLLS problem in Eq. (14). For clarity of explanation, the intensity variation a is not considered. In (a), the nonlinearity is represented as
y=f(x+u)=(2J1(x+u)x+u)2 to simulate the intensity distribution in the Airy disk [32], with J1 the Bessel function of the first kind. The aberration shift the original system y = f (u) horizontally by x = −1.25. The model uncertainty is neglected, i.e., f̂ = f. With single data point P1, the cost function J(x̂) has two minima at x̂ = −1.25 and x̂ = 3.25 as plotted in (b). With points P1 and P2, J(x̂) has one unique global minimum at x̂ = −1.25 but there is a local minimum at x̂ = 2.5. This local minimum vanishes when P3 is added and the domain of convex is increased.

Block diagram of the closed-loop WFSless AO system. The physical WFSless AO system has voltage V(k) as input, but conceptually u(k) can be considered as its input because of the hysteresis compensator and the modal transform.

Accuracy of the neural network model for different number of neurons NQ. VAF increases with NQ in both identification and validation sets for NQ ≤ 20. The difference in VAF is negligible for NQ > 20. Hence 20 neurons are used.

Time line of the WFSless AO system with the MBAC algorithm, including initialization and aberration correction. The initial sampling interval is ts = 20 ms. The computational time tc,1 for the first aberration estimation takes about 40 ms, while the estimation time tc,2 afterwards takes about 20 ms because a better initial guess is provided for the solving the NLLS problem.

Aberration correction with the MBAC+Simplex algorithm and with the simplex algorithm alone, for one static aberration. The MBAC algorithm consists of the initialization and the aberration correction. The initial intensity is 0.17. With the MBAC algorithm, the intensity converges to 0.86 at the 14th time sample, which it takes 30 time samples for the simplex algorithm alone to reach 0.8. The simplex algorithm after MBAC also shows faster convergence than the simplex algorithm alone.

Correction of 20 static aberrations. The initial intensity is 0.49 in average. With the MBAC algorithm, the intensity increases to 0.82 at the 12th time sample and to 0.87 at the 13th time sample. The intensity converges to 0.89 by the MBAC at the 15th time sample, while Simplex 2 needs 45 time samples to reach the same level. The standard deviation of ỹ(k) is also reduced with the MBAC algorithm, indicating that MBAC can give a more deterministic intensity improvement than simplex.